Frequency jumps in the planar vibrations of an elastic beam

The small amplitude transverse vibrations of an elastic beam clamped at both extremities are studied. The beam is modeled as an extensible, shearable planar Kirchhoff elastic rod under large displacements and rotations, and the vibration frequencies are computed both analytically and numerically as a function of the loading. Of particular interest is the variation of mode frequencies as the load is increased through the buckling threshold. While for some modes there is no qualitative changes in the mode frequencies, other modes experience rapid variations after the buckling threshold. For slender beams, these variations become stiffer, eventually resulting in a discontinuous jump of frequency at buckling, in the limit of inextensible, unshearable beams

Elastic Instability behind Brittle Fracture

International audienceWe argue that nucleation of brittle cracks in initially flawless soft elastic solids is preceded by a nonlinear elastic instability, which cannot be captured without accounting for geometrical precise description of finite elastic deformation. As a prototypical problem we consider a homogeneous elastic body subjected to tension and assume that it is weakened by the presence of a free surface which then serves as a site of crack nucleation. We show that in this maximally simplified setting, brittle fracture emerges from a symmetry breaking elastic instability activated by softening and involving large elastic rotations. The implied bifurcation of the homogeneous elastic equilibrium is highly unconventional for nonlinear elasticity as it exhibits an extraordinary sensitivity to geometry, reminiscent of the transition to turbulence in fluids. We trace the post-bifurcational development of this instability beyond the limits of applicability of scale free continuum elasticity and use a phase-field approach to capture the scale dependent sub-continuum strain localization, signaling the formation of actual cracks

A variational model for fracture and debonding of thin films under in-plane loadings

We study fracture and debonding of a thin stiff film bonded to a rigid substrate through a thin compliant layer, introducing a two-dimensional variational fracture model in brittle elasticity. Fractures are naturally distinguished between transverse cracks in the film (curves in 2D) and debonded surfaces (2D planar regions). In order to study the mechanical response of such systems under increasing loads, we formulate a dimension-reduced, rate-independent, irreversible evolution law accounting for both transverse fracture and debonding. We propose a numerical implementation based on a regularized formulation of the fracture problem via a gradient damage functional, and provide an illustration of its capabilities exploring complex crack patterns, showing a qualitative comparison with geometrically involved real life examples. Moreover, we justify the underlying dimension-reduced model in the setting of scalar-valued displacement fields by a rigorous asymptotic analysis using Γ-convergence, starting from the three-dimensional variational fracture (free-discontinuity) problem under precise scaling hypotheses on material and geometric parameters. © 2014 Elsevier Ltd

Foldable structures made of hydrogel bilayers

We discuss self-folding of a thin sheet by using patterned hydrogel bilayers, which act as hinges connecting flat faces. Folding is actuated by heterogeneous swelling due to different crosslinking densities of the polymer network in the two layers. Our analysis is based on a dimensionally reduced plate model, obtained by applying a recently developed theory [1], which provides us with an explicit connection between (three-dimensional) material properties and the curvatures induced at the hinges. This connection offers a recipe for the fabrication and design of the bilayers, by providing the values of the cross-linking density of each layer that need to be imprinted during polymerization in order to produce a desired folded shape upon swelling

Formulation thermodynamique et cadre variationnel pour des modèles d'élastoplasticité à état critique

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Data for: Crack kinking in a variational phase-field model of brittle fracture with strongly anisotropic surface energy

FEniCS based finite element solver for the minimization of the energy functional of a phase-field model for fracture of a material with strongly anisotropic surface energ

Tristability of thin orthotropic shells with uniform initial curvature

Composite shells show a rich multistable behaviour of interest for the design of shape-changing (morphing) structures. Previous studies have investigated how the initial shape determines the shell stability properties. For uniform initial curvatures and orthotropic material behaviour, not more than two stable equilibria have been reported. In this paper, we prove that untwisted, uniformly curved, thin orthotropic shells can have up to three stable equilibrium configurations. Cases of tristability are first documented using a numerical stability analysis of an extensible shallow shell model. Including mid-plane extension shows that the shells must be sufficiently curved in relation to their thickness to be multistable. Thus, an inextensible model allows us to perform an analytical stability analysis. Focusing on untwisted initial configurations, we illustrate with simple analytical results how the material parameters of the shell control the dependence of its multistable behaviour on the initial curvatures. In particular, we show that when the bending stiffness matrix approaches a degeneracy condition, the shell exhibits three stable equilibria for a wide range of initial curvatures. © 2008 The Royal Society